We first analyze the bosonic ground state if the interaction is contact in the \(x\) -\(p_y\)  plane. Note that this is the situation one often encounters in the \(x\) -\(y\)  plane in ultracold atomic gases in the presence of an artificial magnetic field [1]}, [2]}, [3]}, and one expects the formation of vortices. We include the contact interaction in the \(x\) -\(p_y\)  plane by adding a term \(\int dx \int dp_y \tilde{g}_s |\Psi (x,p_y)|^4\)  in the energy functional, where \(\Psi (x,p_y)\)  is the normalized wave function in the \(x\) -\(p_y\)  plane. We choose \(\tilde{g}_s = 10 a_\mathrm {osc} y_0 \omega \) , \(g_s = g_1 = 0\)  and we take all the other parameters to be the same as before; in particular, we take \(\tan \theta = 1\) . In Fig. REF , we plot the ground state obtained by imaginary time evolution of the Gross-Pitaevskii equation. The absolute values of the wave function in the \(x\) -\(y\)  as well as the \(x\) -\(p_y\)  planes are plotted in Figs. REF (a) and REF (b), respectively. We can see two density peaks in the \(x\) -\(y\)  plane, similar to the structure seen in Fig. REF  when the nearest-neighbor interaction is included. The total density of each tube is plotted in Fig. REF (c), where the vertical axis denotes the integral of \(|\Psi _n (x)|^2\)  over \(x\) . The corresponding wave function in the \(x\) -\(p_y\)  plane shows the vortex structure, as confirmed by the phase profile in Fig. REF (d). These vortices, formed by an artificial magnetic field and a contact interaction in the \(x\) -\(p_y\)  plane, are the ordinary vortices which have been observed in ultracold gases in the presence of an artificial magnetic field in real (\(x\) -\(y\) ) space. Comparing these results with the wave function found in Fig. REF , we conclude that the vortex structure found in Fig. REF  has the same origin as the conventional vortices in the presence of an artificial magnetic field.
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